Glossary

29.7 Probability: The Heisenberg Uncertainty Principle

3 min read•june 18, 2024

Quantum uncertainty challenges our everyday understanding of reality. It reveals that particles can't have precise positions and momenta simultaneously, and that measuring one affects the other. This fundamental principle shapes the weird, probabilistic nature of the quantum world.

Heisenberg's uncertainty principle has far-reaching implications. It impacts how we understand particle behavior, energy states, and even the act of measurement itself. These concepts form the foundation of modern and its mind-bending interpretations.

Quantum Uncertainty

Heisenberg's uncertainty principle applications

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  • States the product of uncertainties in (Δx\Delta x) and (Δp\Delta p) is always greater than or equal to h4π\frac{h}{4\pi}
    • Mathematically expressed as ΔxΔp≥h4π\Delta x \Delta p \geq \frac{h}{4\pi}, where hh represents
  • Implies the more precisely a particle's is known, the less precisely its momentum can be determined, and vice versa
  • example: If an electron's position is measured with an uncertainty of 1 nm, the uncertainty in its momentum must be at least h4π⋅1 nm\frac{h}{4\pi \cdot 1 \text{ nm}}
  • Photon example: If a photon's is measured with an uncertainty of 1 nm, the uncertainty in its momentum must be at least h4π⋅1 nm\frac{h}{4\pi \cdot 1 \text{ nm}}
  • The principle is a fundamental aspect of , describing the dual nature of quantum entities

Measurement effects in quantum systems

  • In , the act of measurement can disturb the system being measured, unlike in classical physics where measurement is assumed to have no effect
  • Measuring a particle's position or momentum collapses its , forcing it into a definite state from a previous of multiple states
  • The collapse of the wavefunction due to measurement is a fundamental aspect of quantum mechanics
    • Leads to the probabilistic nature of quantum phenomena (, )
  • example: Measuring which slit an electron passes through destroys the interference pattern, demonstrating the effect of measurement on quantum systems
  • This phenomenon is known as the , where the act of observation influences the system being observed

Energy-time uncertainty for particle analysis

  • Another form of Heisenberg's uncertainty principle relating uncertainties in energy (ΔE\Delta E) and time (Δt\Delta t)
    • Mathematically expressed as ΔEΔt≥h4π\Delta E \Delta t \geq \frac{h}{4\pi}
  • Has important implications for short-lived particles and
    • A particle with a short lifetime (Δt\Delta t) must have a large uncertainty in its energy (ΔE\Delta E)
    • Excited states with short lifetimes have broad energy levels, while long-lived states have narrow energy levels
  • Atomic excited state example: If the lifetime of an excited atomic state is 1 ns, the uncertainty in the energy of this state must be at least h4π⋅1 ns\frac{h}{4\pi \cdot 1 \text{ ns}}
  • example: Muons, with a lifetime of 2.2 µs, have an energy uncertainty of at least h4π⋅2.2 µs\frac{h}{4\pi \cdot 2.2 \text{ µs}}, which allows them to decay into electrons and

Interpretations and Mathematical Formulations

  • The , developed by Niels Bohr and Werner Heisenberg, provides a framework for understanding quantum mechanics and the uncertainty principle
  • Quantum states are described by probability amplitudes, which are complex-valued functions that determine the likelihood of measurement outcomes
  • , developed by Heisenberg, Born, and Jordan, offers a mathematical formulation of quantum mechanics that inherently incorporates the uncertainty principle

Key Terms to Review (34)

Antielectron: An antielectron, also known as a positron, is the antimatter counterpart of an electron. It has the same mass as an electron but carries a positive charge.
Copenhagen interpretation: The Copenhagen interpretation is a fundamental explanation of quantum mechanics that asserts that particles exhibit both wave and particle characteristics, and that the act of measurement affects the system being observed. It emphasizes the role of probability in quantum events, indicating that outcomes are not determined until they are measured, which ties into concepts like the Heisenberg Uncertainty Principle.
De Broglie wavelength: The de Broglie wavelength is the wavelength associated with a particle and is inversely proportional to its momentum. It highlights the wave-particle duality of matter.
Double-Slit Experiment: The double-slit experiment is a fundamental experiment in quantum physics that demonstrates the wave-particle duality of light and other quantum particles. It involves the passage of a beam of light or particles through two narrow slits, resulting in an interference pattern that reveals the wave-like behavior of the system.
Electron: An electron is a fundamental subatomic particle that carries a negative electric charge and is found in all atoms, playing a crucial role in various physical and chemical phenomena. Electrons are responsible for the flow of electric current, the formation of chemical bonds, and the behavior of matter at the atomic and molecular levels. The concept of the electron is central to understanding topics such as static electricity, electric fields, magnetic fields, the photoelectric effect, quantum mechanics, and the structure of atoms. Electrons are the building blocks of matter and are essential for understanding the fundamental nature of the universe.
Electron spin: Electron spin is a fundamental property of electrons that describes their intrinsic angular momentum, which can take on one of two values: 'up' or 'down'. This concept is crucial in quantum mechanics as it plays a significant role in determining the magnetic properties of materials and the arrangement of electrons in atoms. Electron spin contributes to the behavior of particles in quantum systems and influences the uncertainty of their position and momentum as outlined by the Heisenberg Uncertainty Principle.
Energy-time uncertainty: Energy-time uncertainty is a concept in quantum mechanics that describes the inherent limitations in simultaneously knowing the exact energy of a system and the time duration during which that energy is defined. This principle indicates that the more precisely one can measure the energy of a quantum state, the less precisely one can know the time over which that state exists, and vice versa. This relationship is a manifestation of the Heisenberg Uncertainty Principle, highlighting the fundamental probabilistic nature of quantum systems.
Excited States: Excited states refer to the higher energy levels that an atom or molecule can occupy when electrons are promoted to orbitals with greater principal quantum numbers. These higher energy states are less stable than the ground state and the electrons will eventually return to the lower energy ground state, often releasing energy in the form of photons.
Heisenberg uncertainty principle: The Heisenberg uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. This inherent limitation arises due to the wave-particle duality of quantum objects.
Heisenberg Uncertainty Principle: The Heisenberg Uncertainty Principle states that the more precisely the position of a particle is determined, the less precisely its momentum can be known, and vice versa. This fundamental principle of quantum mechanics places a limit on the accuracy with which certain pairs of physical properties, such as position and momentum, can be simultaneously measured.
Heisenberg’s uncertainty principle: Heisenberg's uncertainty principle states that it is impossible to simultaneously determine both the position and momentum of a particle with absolute precision. The more accurately one of these properties is measured, the less accurately the other can be known.
Matrix mechanics: Matrix mechanics is a formulation of quantum mechanics that represents physical quantities as matrices and employs matrix algebra to describe the behavior of quantum systems. This approach was one of the first complete formulations of quantum mechanics and laid the groundwork for understanding the probabilistic nature of quantum states, connecting directly to the Heisenberg Uncertainty Principle.
Momentum: Momentum is a vector quantity that represents the product of an object's mass and velocity. It is a measure of an object's quantity of motion and is conserved in a closed system, meaning the total momentum of a system remains constant unless acted upon by an external force.
Muon: The muon is a subatomic particle that is similar to the electron but with a much greater mass. It is an unstable particle that is part of the lepton family and plays a crucial role in understanding the Heisenberg Uncertainty Principle and the behavior of particles in high-energy physics.
Muon family number: The muon family number is a quantum number assigned to leptons, specifically muons and their associated neutrinos, to represent the conservation of lepton family type in particle interactions. It is denoted by $L_\mu$.
Neutrinos: Neutrinos are electrically neutral, weakly interacting elementary particles that travel at or near the speed of light. They are produced in nuclear reactions and play a crucial role in understanding the Heisenberg Uncertainty Principle and the probabilistic nature of quantum mechanics.
Observer Effect: The observer effect refers to the phenomenon where the act of observing a system or phenomenon changes the behavior or characteristics of that system. This concept is particularly important in the context of quantum mechanics and the Heisenberg Uncertainty Principle.
Photon Polarization: Photon polarization refers to the orientation of the electric field vector of a photon, which describes the direction of the electromagnetic wave's oscillation. This property of light is important in understanding the wave-particle duality of photons and their interactions with matter, particularly in the context of the Heisenberg Uncertainty Principle.
Planck's constant: Planck's constant is a fundamental physical constant that represents the smallest possible change in energy or action. It is a crucial parameter in quantum mechanics and is denoted by the symbol 'h'. Planck's constant establishes the relationship between the energy of a photon and its frequency, and it is a key factor in understanding the quantization of energy and the wave-particle duality of matter and energy.
Position: Position refers to the specific point in space that an object occupies, typically described using coordinates relative to a reference point. It is a vector quantity that has both magnitude and direction.
Position: Position refers to the location of an object or a particle in space, typically described using a coordinate system. It is a fundamental concept in physics that is crucial for understanding various topics, including displacement, motion, and the Heisenberg Uncertainty Principle.
Probability Amplitude: Probability amplitude is a complex-valued function that describes the quantum mechanical state of an object. It is a fundamental concept in quantum mechanics that represents the probability of finding a particle in a particular state or location.
Probability distribution: A probability distribution describes how probabilities are distributed over the possible outcomes of a random variable. In quantum physics, it represents the likelihood of finding a particle in various states or positions.
Quantum mechanics: Quantum mechanics is a fundamental theory in physics that describes the behavior of particles at atomic and subatomic scales. It explains phenomena that cannot be accounted for by classical physics.
Quantum Mechanics: Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy on the atomic and subatomic scale. It is a powerful framework for understanding the properties and interactions of particles at the quantum level, which are often counterintuitive and defy classical physics.
Superposition: Superposition is the principle that when two or more waves overlap, the resulting wave displacement is the sum of the individual wave displacements. This principle applies to all types of waves, including mechanical and electromagnetic.
Superposition: Superposition is the principle that when two or more waves or oscillations occur at the same time and location, their net displacement is the vector sum of the individual displacements. This concept is fundamental to the understanding of various wave phenomena, including interference and diffraction.
Uncertainty in energy: Uncertainty in energy is a fundamental concept in quantum mechanics, highlighting the inherent limitations in simultaneously knowing the exact energy and time of a particle. It is quantitatively expressed by the Heisenberg Uncertainty Principle.
Uncertainty in momentum: Uncertainty in momentum refers to the limitation in determining the exact momentum of a particle due to inherent quantum fluctuations. This concept is a fundamental aspect of the Heisenberg Uncertainty Principle, which states that position and momentum cannot both be precisely measured simultaneously.
Uncertainty in position: Uncertainty in position describes the inherent limitation in simultaneously knowing both the exact position and momentum of a particle. This concept is a fundamental aspect of quantum mechanics.
Uncertainty in time: Uncertainty in time is a concept in quantum physics that describes the inability to precisely measure both the exact time and energy of a particle simultaneously. This stems from the Heisenberg Uncertainty Principle which limits the precision of such measurements.
Wave-Particle Duality: Wave-particle duality is a fundamental concept in quantum physics that describes the dual nature of light and matter, where they exhibit characteristics of both waves and particles depending on the context and experimental conditions. This principle is central to understanding the behavior of electromagnetic radiation and the properties of subatomic particles.
Wavefunction: The wavefunction is a mathematical function that describes the quantum state of an object or particle. It is a fundamental concept in quantum mechanics that provides a complete description of the behavior and properties of a particle or system.
Wavelength: Wavelength is a fundamental characteristic of waves, representing the distance between consecutive peaks or troughs in a wave. It is a crucial parameter that describes the spatial extent of a wave and is closely related to other wave properties such as frequency and speed.
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